At a certain filling station, customers arrive in a Poisson process with an average time of 12 per hour. The time interval between service follows exponential distribution and as such the mean time taken to service to a unit is 2 minutes. Evaluate:
i) Probability that there is no customer at the counter.
ii) Probability that there are more than two customers at the counter.
iii) Average number of customers in a queue waiting for service.
iv) Expected waiting time of a customer in the system.
v) Probability that a customer wait for 0.11 minutes in a queue.
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Consider a Markov chain with transition probability matrix
i) Whether the chain is irreducible? If irreducible classify the states of a Markov chain i.e., recurrent, transient, periodic and mean recurrence time.
ii) Find the limiting probability vector.
b) If N1 ,(t), N2 (t) are two independent Poisson process with parameters λ1 and λ2 respectively, then show that
A service station has 5 mechanics each of whom can service a scooter in 2 hours on the average. The scooters are registered at a single counter and then sent for servicing to different mechanics. Scooters arrive at a service station at an average rate of 2 scooters per hour. Assuming that the scooter arrivals are Poisson and service times are exponentially distributed, determine:
i) Identify the model.
ii) The probability that the system shall be idle.
iii) The probability that there shall be 3 scooters in the service centre.
iv) The expected number of scooters waiting in a queue.
v) The expected number of scooters in the service centre.
vi) The average waiting time in a queue.
Let ( ,X Y) have the joint p.d.f. given by:
f(x,y)=
i) Find the marginal p.d.f.’s of X and Y.
ii) Test the independence of X and Y.
iii) Find the conditional distribution of X given Y = .y
iv) Compute (E X | Y = )y and (E Y | X = ).x
At a certain filling station, customers arrive in a Poisson process with an average time of 12 per hour. The time interval between service follows exponential distribution and as such the mean time taken to service to a unit is 2 minutes. Evaluate:
i) Probability that there is no customer at the counter.
ii) Probability that there are more than two customers at the counter.
iii) Average number of customers in a queue waiting for service.
iv) Expected waiting time of a customer in the system.
v) Probability that a customer wait for 0.11 minutes in a queue.
A box contains two coins: a regular coin and one fake two-headed coin. One coin is chosen at random and tossed twice. The following events are defined: (5)
A: first coin toss results in a head.
B: second coin toss results in a head.
C: coin 1 (regular) has been selected.
Find (P C|A ), (P |B C), (P A ∩ C|)B ), (P A), (P )B and (P A∩ ).B
A particular component in a machine is replaced instantaneously on failure. The successive component lifetimes are uniformly distributed over the interval ]5,2[ years. Further, planned replacements take place every 3 years.
i) long-terms rate of replacements.
ii) long-terms rate of failures. Compute